LMFDB - Embedded newform 6000.2.a.bi.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6000,2,Mod(1,6000)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6000, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6000.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6000 = 2^{4} \cdot 3 \cdot 5^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6000.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$47.9102412128$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.13025.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 12x^{2} + 3x + 29 $ x^4 - x^3 - 12x^2 + 3x + 29
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3000)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$3.50578$ of defining polynomial
Character	$\chi$	$=$	6000.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +4.05444 q^{7} +1.00000 q^{9} +1.33909 q^{11} -7.12382 q^{13} -3.15746 q^{17} -4.78473 q^{19} +4.05444 q^{21} -0.103022 q^{23} +1.00000 q^{27} -6.22113 q^{29} +0.402761 q^{31} +1.33909 q^{33} -0.327525 q^{37} -7.12382 q^{39} -7.62389 q^{41} -9.18749 q^{43} -10.8993 q^{47} +9.43849 q^{49} -3.15746 q^{51} -0.730286 q^{53} -4.78473 q^{57} -8.39353 q^{59} -3.65376 q^{61} +4.05444 q^{63} +3.90854 q^{67} -0.103022 q^{69} +7.08680 q^{71} -2.86904 q^{73} +5.42926 q^{77} -12.4480 q^{79} +1.00000 q^{81} +3.96869 q^{83} -6.22113 q^{87} +18.0624 q^{89} -28.8831 q^{91} +0.402761 q^{93} +14.6355 q^{97} +1.33909 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} + q^{7} + 4 q^{9} - q^{11} - 11 q^{13} - 8 q^{19} + q^{21} - 3 q^{23} + 4 q^{27} - 3 q^{29} - 14 q^{31} - q^{33} - 21 q^{37} - 11 q^{39} + 7 q^{41} - 10 q^{43} - 9 q^{47} + 15 q^{49} - 7 q^{53}+ \cdots - q^{99}+O(q^{100}) $ 4 * q + 4 * q^3 + q^7 + 4 * q^9 - q^11 - 11 * q^13 - 8 * q^19 + q^21 - 3 * q^23 + 4 * q^27 - 3 * q^29 - 14 * q^31 - q^33 - 21 * q^37 - 11 * q^39 + 7 * q^41 - 10 * q^43 - 9 * q^47 + 15 * q^49 - 7 * q^53 - 8 * q^57 - 12 * q^59 - 3 * q^61 + q^63 - 13 * q^67 - 3 * q^69 - 33 * q^71 - 11 * q^73 - 7 * q^77 - 13 * q^79 + 4 * q^81 - 21 * q^83 - 3 * q^87 + 12 * q^89 - 17 * q^91 - 14 * q^93 - 5 * q^97 - q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.05444	1.53243	0.766217	−	0.642582i	$-0.222137\pi$
$7$	4.05444	1.53243	0.766217	+	0.642582i	$0.222137\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.33909	0.403751	0.201875	−	0.979411i	$-0.435296\pi$
$11$	1.33909	0.403751	0.201875	+	0.979411i	$0.435296\pi$
$12$	0	0
$13$	−7.12382	−1.97579	−0.987896	−	0.155120i	$-0.950423\pi$
$13$	−7.12382	−1.97579	−0.987896	+	0.155120i	$0.950423\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−3.15746	−0.765797	−0.382899	−	0.923790i	$-0.625074\pi$
$17$	−3.15746	−0.765797	−0.382899	+	0.923790i	$0.625074\pi$
$18$	0	0
$19$	−4.78473	−1.09769	−0.548846	−	0.835924i	$-0.684933\pi$
$19$	−4.78473	−1.09769	−0.548846	+	0.835924i	$0.684933\pi$
$20$	0	0
$21$	4.05444	0.884752
$22$	0	0
$23$	−0.103022	−0.0214815	−0.0107408	−	0.999942i	$-0.503419\pi$
$23$	−0.103022	−0.0214815	−0.0107408	+	0.999942i	$0.503419\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−6.22113	−1.15524	−0.577618	−	0.816307i	$-0.696018\pi$
$29$	−6.22113	−1.15524	−0.577618	+	0.816307i	$0.696018\pi$
$30$	0	0
$31$	0.402761	0.0723379	0.0361690	−	0.999346i	$-0.488485\pi$
$31$	0.402761	0.0723379	0.0361690	+	0.999346i	$0.488485\pi$
$32$	0	0
$33$	1.33909	0.233106
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−0.327525	−0.0538448	−0.0269224	−	0.999638i	$-0.508571\pi$
$37$	−0.327525	−0.0538448	−0.0269224	+	0.999638i	$0.508571\pi$
$38$	0	0
$39$	−7.12382	−1.14072
$40$	0	0
$41$	−7.62389	−1.19065	−0.595326	−	0.803484i	$-0.702977\pi$
$41$	−7.62389	−1.19065	−0.595326	+	0.803484i	$0.702977\pi$
$42$	0	0
$43$	−9.18749	−1.40108	−0.700539	−	0.713614i	$-0.747057\pi$
$43$	−9.18749	−1.40108	−0.700539	+	0.713614i	$0.747057\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−10.8993	−1.58983	−0.794914	−	0.606722i	$-0.792484\pi$
$47$	−10.8993	−1.58983	−0.794914	+	0.606722i	$0.792484\pi$
$48$	0	0
$49$	9.43849	1.34836
$50$	0	0
$51$	−3.15746	−0.442133
$52$	0	0
$53$	−0.730286	−0.100312	−0.0501562	−	0.998741i	$-0.515972\pi$
$53$	−0.730286	−0.100312	−0.0501562	+	0.998741i	$0.515972\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−4.78473	−0.633753
$58$	0	0
$59$	−8.39353	−1.09274	−0.546372	−	0.837542i	$-0.683992\pi$
$59$	−8.39353	−1.09274	−0.546372	+	0.837542i	$0.683992\pi$
$60$	0	0
$61$	−3.65376	−0.467816	−0.233908	−	0.972259i	$-0.575152\pi$
$61$	−3.65376	−0.467816	−0.233908	+	0.972259i	$0.575152\pi$
$62$	0	0
$63$	4.05444	0.510812
$64$	0	0
$65$	0	0
$66$	0	0
$67$	3.90854	0.477504	0.238752	−	0.971081i	$-0.423262\pi$
$67$	3.90854	0.477504	0.238752	+	0.971081i	$0.423262\pi$
$68$	0	0
$69$	−0.103022	−0.0124024
$70$	0	0
$71$	7.08680	0.841048	0.420524	−	0.907281i	$-0.361846\pi$
$71$	7.08680	0.841048	0.420524	+	0.907281i	$0.361846\pi$
$72$	0	0
$73$	−2.86904	−0.335795	−0.167898	−	0.985804i	$-0.553698\pi$
$73$	−2.86904	−0.335795	−0.167898	+	0.985804i	$0.553698\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	5.42926	0.618722
$78$	0	0
$79$	−12.4480	−1.40051	−0.700253	−	0.713895i	$-0.746930\pi$
$79$	−12.4480	−1.40051	−0.700253	+	0.713895i	$0.746930\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	3.96869	0.435620	0.217810	−	0.975991i	$-0.430109\pi$
$83$	3.96869	0.435620	0.217810	+	0.975991i	$0.430109\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−6.22113	−0.666975
$88$	0	0
$89$	18.0624	1.91461	0.957304	−	0.289082i	$-0.0933499\pi$
$89$	18.0624	1.91461	0.957304	+	0.289082i	$0.0933499\pi$
$90$	0	0
$91$	−28.8831	−3.02777
$92$	0	0
$93$	0.402761	0.0417643
$94$	0	0
$95$	0	0
$96$	0	0
$97$	14.6355	1.48601	0.743003	−	0.669288i	$-0.233401\pi$
$97$	14.6355	1.48601	0.743003	+	0.669288i	$0.233401\pi$
$98$	0	0
$99$	1.33909	0.134584
$100$	0	0
$101$	−7.35988	−0.732336	−0.366168	−	0.930549i	$-0.619330\pi$
$101$	−7.35988	−0.732336	−0.366168	+	0.930549i	$0.619330\pi$
$102$	0	0
$103$	−3.88060	−0.382367	−0.191183	−	0.981554i	$-0.561233\pi$
$103$	−3.88060	−0.382367	−0.191183	+	0.981554i	$0.561233\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−9.39706	−0.908448	−0.454224	−	0.890888i	$-0.650083\pi$
$107$	−9.39706	−0.908448	−0.454224	+	0.890888i	$0.650083\pi$
$108$	0	0
$109$	−4.89361	−0.468723	−0.234361	−	0.972150i	$-0.575300\pi$
$109$	−4.89361	−0.468723	−0.234361	+	0.972150i	$0.575300\pi$
$110$	0	0
$111$	−0.327525	−0.0310873
$112$	0	0
$113$	−16.8436	−1.58451	−0.792256	−	0.610189i	$-0.791093\pi$
$113$	−16.8436	−1.58451	−0.792256	+	0.610189i	$0.791093\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−7.12382	−0.658597
$118$	0	0
$119$	−12.8017	−1.17353
$120$	0	0
$121$	−9.20684	−0.836985
$122$	0	0
$123$	−7.62389	−0.687423
$124$	0	0
$125$	0	0
$126$	0	0
$127$	17.1711	1.52369	0.761845	−	0.647760i	$-0.224294\pi$
$127$	17.1711	1.52369	0.761845	+	0.647760i	$0.224294\pi$
$128$	0	0
$129$	−9.18749	−0.808913
$130$	0	0
$131$	−6.34624	−0.554473	−0.277237	−	0.960802i	$-0.589419\pi$
$131$	−6.34624	−0.554473	−0.277237	+	0.960802i	$0.589419\pi$
$132$	0	0
$133$	−19.3994	−1.68214
$134$	0	0
$135$	0	0
$136$	0	0
$137$	18.9615	1.61999	0.809997	−	0.586434i	$-0.199469\pi$
$137$	18.9615	1.61999	0.809997	+	0.586434i	$0.199469\pi$
$138$	0	0
$139$	15.1412	1.28426	0.642132	−	0.766594i	$-0.278050\pi$
$139$	15.1412	1.28426	0.642132	+	0.766594i	$0.278050\pi$
$140$	0	0
$141$	−10.8993	−0.917888
$142$	0	0
$143$	−9.53943	−0.797727
$144$	0	0
$145$	0	0
$146$	0	0
$147$	9.43849	0.778474
$148$	0	0
$149$	23.8194	1.95136	0.975681	−	0.219193i	$-0.0703425\pi$
$149$	23.8194	1.95136	0.975681	+	0.219193i	$0.0703425\pi$
$150$	0	0
$151$	20.1469	1.63954	0.819768	−	0.572696i	$-0.194103\pi$
$151$	20.1469	1.63954	0.819768	+	0.572696i	$0.194103\pi$
$152$	0	0
$153$	−3.15746	−0.255266
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−6.15513	−0.491233	−0.245616	−	0.969367i	$-0.578990\pi$
$157$	−6.15513	−0.491233	−0.245616	+	0.969367i	$0.578990\pi$
$158$	0	0
$159$	−0.730286	−0.0579154
$160$	0	0
$161$	−0.417695	−0.0329190
$162$	0	0
$163$	−8.16525	−0.639552	−0.319776	−	0.947493i	$-0.603608\pi$
$163$	−8.16525	−0.639552	−0.319776	+	0.947493i	$0.603608\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−10.6401	−0.823357	−0.411678	−	0.911329i	$-0.635057\pi$
$167$	−10.6401	−0.823357	−0.411678	+	0.911329i	$0.635057\pi$
$168$	0	0
$169$	37.7488	2.90375
$170$	0	0
$171$	−4.78473	−0.365897
$172$	0	0
$173$	6.66116	0.506439	0.253219	−	0.967409i	$-0.418511\pi$
$173$	6.66116	0.506439	0.253219	+	0.967409i	$0.418511\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−8.39353	−0.630896
$178$	0	0
$179$	−1.77316	−0.132532	−0.0662662	−	0.997802i	$-0.521109\pi$
$179$	−1.77316	−0.132532	−0.0662662	+	0.997802i	$0.521109\pi$
$180$	0	0
$181$	16.9027	1.25637	0.628183	−	0.778065i	$-0.283799\pi$
$181$	16.9027	1.25637	0.628183	+	0.778065i	$0.283799\pi$
$182$	0	0
$183$	−3.65376	−0.270094
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−4.22813	−0.309191
$188$	0	0
$189$	4.05444	0.294917
$190$	0	0
$191$	−8.78473	−0.635641	−0.317820	−	0.948151i	$-0.602951\pi$
$191$	−8.78473	−0.635641	−0.317820	+	0.948151i	$0.602951\pi$
$192$	0	0
$193$	12.4745	0.897932	0.448966	−	0.893549i	$-0.351792\pi$
$193$	12.4745	0.897932	0.448966	+	0.893549i	$0.351792\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−16.2500	−1.15776	−0.578881	−	0.815412i	$-0.696510\pi$
$197$	−16.2500	−1.15776	−0.578881	+	0.815412i	$0.696510\pi$
$198$	0	0
$199$	15.6334	1.10822	0.554110	−	0.832443i	$-0.313059\pi$
$199$	15.6334	1.10822	0.554110	+	0.832443i	$0.313059\pi$
$200$	0	0
$201$	3.90854	0.275687
$202$	0	0
$203$	−25.2232	−1.77032
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−0.103022	−0.00716050
$208$	0	0
$209$	−6.40718	−0.443194
$210$	0	0
$211$	23.1389	1.59295	0.796474	−	0.604673i	$-0.206696\pi$
$211$	23.1389	1.59295	0.796474	+	0.604673i	$0.206696\pi$
$212$	0	0
$213$	7.08680	0.485580
$214$	0	0
$215$	0	0
$216$	0	0
$217$	1.63297	0.110853
$218$	0	0
$219$	−2.86904	−0.193872
$220$	0	0
$221$	22.4932	1.51306
$222$	0	0
$223$	−16.9651	−1.13606	−0.568032	−	0.823006i	$-0.692295\pi$
$223$	−16.9651	−1.13606	−0.568032	+	0.823006i	$0.692295\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	10.7757	0.715205	0.357603	−	0.933874i	$-0.383594\pi$
$227$	10.7757	0.715205	0.357603	+	0.933874i	$0.383594\pi$
$228$	0	0
$229$	18.5696	1.22711	0.613557	−	0.789650i	$-0.289738\pi$
$229$	18.5696	1.22711	0.613557	+	0.789650i	$0.289738\pi$
$230$	0	0
$231$	5.42926	0.357219
$232$	0	0
$233$	−1.65039	−0.108121	−0.0540604	−	0.998538i	$-0.517216\pi$
$233$	−1.65039	−0.108121	−0.0540604	+	0.998538i	$0.517216\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−12.4480	−0.808583
$238$	0	0
$239$	−19.0660	−1.23328	−0.616639	−	0.787246i	$-0.711506\pi$
$239$	−19.0660	−1.23328	−0.616639	+	0.787246i	$0.711506\pi$
$240$	0	0
$241$	15.5752	1.00328	0.501642	−	0.865075i	$-0.332729\pi$
$241$	15.5752	1.00328	0.501642	+	0.865075i	$0.332729\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	34.0855	2.16881
$248$	0	0
$249$	3.96869	0.251505
$250$	0	0
$251$	−18.2130	−1.14959	−0.574796	−	0.818297i	$-0.694918\pi$
$251$	−18.2130	−1.14959	−0.574796	+	0.818297i	$0.694918\pi$
$252$	0	0
$253$	−0.137955	−0.00867317
$254$	0	0
$255$	0	0
$256$	0	0
$257$	14.8505	0.926348	0.463174	−	0.886267i	$-0.346711\pi$
$257$	14.8505	0.926348	0.463174	+	0.886267i	$0.346711\pi$
$258$	0	0
$259$	−1.32793	−0.0825137
$260$	0	0
$261$	−6.22113	−0.385078
$262$	0	0
$263$	−28.2291	−1.74068	−0.870340	−	0.492452i	$-0.836101\pi$
$263$	−28.2291	−1.74068	−0.870340	+	0.492452i	$0.836101\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	18.0624	1.10540
$268$	0	0
$269$	−2.33130	−0.142142	−0.0710710	−	0.997471i	$-0.522642\pi$
$269$	−2.33130	−0.142142	−0.0710710	+	0.997471i	$0.522642\pi$
$270$	0	0
$271$	9.84503	0.598043	0.299021	−	0.954246i	$-0.403340\pi$
$271$	9.84503	0.598043	0.299021	+	0.954246i	$0.403340\pi$
$272$	0	0
$273$	−28.8831	−1.74808
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−24.4616	−1.46976	−0.734878	−	0.678199i	$-0.762761\pi$
$277$	−24.4616	−1.46976	−0.734878	+	0.678199i	$0.762761\pi$
$278$	0	0
$279$	0.402761	0.0241126
$280$	0	0
$281$	15.6447	0.933284	0.466642	−	0.884446i	$-0.345464\pi$
$281$	15.6447	0.933284	0.466642	+	0.884446i	$0.345464\pi$
$282$	0	0
$283$	9.82969	0.584314	0.292157	−	0.956370i	$-0.405627\pi$
$283$	9.82969	0.584314	0.292157	+	0.956370i	$0.405627\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−30.9106	−1.82460
$288$	0	0
$289$	−7.03043	−0.413555
$290$	0	0
$291$	14.6355	0.857946
$292$	0	0
$293$	−17.3449	−1.01330	−0.506651	−	0.862151i	$-0.669117\pi$
$293$	−17.3449	−1.01330	−0.506651	+	0.862151i	$0.669117\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	1.33909	0.0777019
$298$	0	0
$299$	0.733907	0.0424430
$300$	0	0
$301$	−37.2501	−2.14706
$302$	0	0
$303$	−7.35988	−0.422814
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−22.1862	−1.26623	−0.633117	−	0.774056i	$-0.718225\pi$
$307$	−22.1862	−1.26623	−0.633117	+	0.774056i	$0.718225\pi$
$308$	0	0
$309$	−3.88060	−0.220760
$310$	0	0
$311$	19.9407	1.13073	0.565365	−	0.824841i	$-0.308735\pi$
$311$	19.9407	1.13073	0.565365	+	0.824841i	$0.308735\pi$
$312$	0	0
$313$	10.9945	0.621449	0.310724	−	0.950500i	$-0.399428\pi$
$313$	10.9945	0.621449	0.310724	+	0.950500i	$0.399428\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	24.6249	1.38307	0.691537	−	0.722341i	$-0.256934\pi$
$317$	24.6249	1.38307	0.691537	+	0.722341i	$0.256934\pi$
$318$	0	0
$319$	−8.33066	−0.466427
$320$	0	0
$321$	−9.39706	−0.524493
$322$	0	0
$323$	15.1076	0.840609
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−4.89361	−0.270617
$328$	0	0
$329$	−44.1906	−2.43631
$330$	0	0
$331$	−5.98481	−0.328955	−0.164478	−	0.986381i	$-0.552594\pi$
$331$	−5.98481	−0.328955	−0.164478	+	0.986381i	$0.552594\pi$
$332$	0	0
$333$	−0.327525	−0.0179483
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−21.3564	−1.16335	−0.581677	−	0.813420i	$-0.697603\pi$
$337$	−21.3564	−1.16335	−0.581677	+	0.813420i	$0.697603\pi$
$338$	0	0
$339$	−16.8436	−0.914818
$340$	0	0
$341$	0.539332	0.0292065
$342$	0	0
$343$	9.88671	0.533832
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−9.13096	−0.490176	−0.245088	−	0.969501i	$-0.578817\pi$
$347$	−9.13096	−0.490176	−0.245088	+	0.969501i	$0.578817\pi$
$348$	0	0
$349$	14.9106	0.798147	0.399074	−	0.916919i	$-0.369332\pi$
$349$	14.9106	0.798147	0.399074	+	0.916919i	$0.369332\pi$
$350$	0	0
$351$	−7.12382	−0.380241
$352$	0	0
$353$	−15.7519	−0.838388	−0.419194	−	0.907897i	$-0.637687\pi$
$353$	−15.7519	−0.838388	−0.419194	+	0.907897i	$0.637687\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−12.8017	−0.677540
$358$	0	0
$359$	17.4164	0.919203	0.459601	−	0.888125i	$-0.347992\pi$
$359$	17.4164	0.919203	0.459601	+	0.888125i	$0.347992\pi$
$360$	0	0
$361$	3.89361	0.204927
$362$	0	0
$363$	−9.20684	−0.483234
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−10.5432	−0.550351	−0.275175	−	0.961394i	$-0.588736\pi$
$367$	−10.5432	−0.550351	−0.275175	+	0.961394i	$0.588736\pi$
$368$	0	0
$369$	−7.62389	−0.396884
$370$	0	0
$371$	−2.96090	−0.153722
$372$	0	0
$373$	−36.8381	−1.90741	−0.953703	−	0.300750i	$-0.902763\pi$
$373$	−36.8381	−1.90741	−0.953703	+	0.300750i	$0.902763\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	44.3182	2.28250
$378$	0	0
$379$	32.4151	1.66505	0.832526	−	0.553985i	$-0.186894\pi$
$379$	32.4151	1.66505	0.832526	+	0.553985i	$0.186894\pi$
$380$	0	0
$381$	17.1711	0.879703
$382$	0	0
$383$	−13.0618	−0.667429	−0.333714	−	0.942674i	$-0.608302\pi$
$383$	−13.0618	−0.667429	−0.333714	+	0.942674i	$0.608302\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−9.18749	−0.467026
$388$	0	0
$389$	−8.43279	−0.427559	−0.213780	−	0.976882i	$-0.568577\pi$
$389$	−8.43279	−0.427559	−0.213780	+	0.976882i	$0.568577\pi$
$390$	0	0
$391$	0.325287	0.0164505
$392$	0	0
$393$	−6.34624	−0.320125
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−6.21309	−0.311826	−0.155913	−	0.987771i	$-0.549832\pi$
$397$	−6.21309	−0.311826	−0.155913	+	0.987771i	$0.549832\pi$
$398$	0	0
$399$	−19.3994	−0.971184
$400$	0	0
$401$	1.29845	0.0648416	0.0324208	−	0.999474i	$-0.489678\pi$
$401$	1.29845	0.0648416	0.0324208	+	0.999474i	$0.489678\pi$
$402$	0	0
$403$	−2.86919	−0.142925
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−0.438586	−0.0217399
$408$	0	0
$409$	−13.5402	−0.669521	−0.334760	−	0.942303i	$-0.608655\pi$
$409$	−13.5402	−0.669521	−0.334760	+	0.942303i	$0.608655\pi$
$410$	0	0
$411$	18.9615	0.935304
$412$	0	0
$413$	−34.0311	−1.67456
$414$	0	0
$415$	0	0
$416$	0	0
$417$	15.1412	0.741470
$418$	0	0
$419$	−0.547371	−0.0267408	−0.0133704	−	0.999911i	$-0.504256\pi$
$419$	−0.547371	−0.0267408	−0.0133704	+	0.999911i	$0.504256\pi$
$420$	0	0
$421$	−0.319091	−0.0155515	−0.00777577	−	0.999970i	$-0.502475\pi$
$421$	−0.319091	−0.0155515	−0.00777577	+	0.999970i	$0.502475\pi$
$422$	0	0
$423$	−10.8993	−0.529943
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−14.8140	−0.716898
$428$	0	0
$429$	−9.53943	−0.460568
$430$	0	0
$431$	−34.4929	−1.66147	−0.830733	−	0.556671i	$-0.812078\pi$
$431$	−34.4929	−1.66147	−0.830733	+	0.556671i	$0.812078\pi$
$432$	0	0
$433$	24.6600	1.18509	0.592543	−	0.805539i	$-0.298124\pi$
$433$	24.6600	1.18509	0.592543	+	0.805539i	$0.298124\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0.492930	0.0235801
$438$	0	0
$439$	−27.1693	−1.29672	−0.648361	−	0.761333i	$-0.724545\pi$
$439$	−27.1693	−1.29672	−0.648361	+	0.761333i	$0.724545\pi$
$440$	0	0
$441$	9.43849	0.449452
$442$	0	0
$443$	7.75124	0.368272	0.184136	−	0.982901i	$-0.441051\pi$
$443$	7.75124	0.368272	0.184136	+	0.982901i	$0.441051\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	23.8194	1.12662
$448$	0	0
$449$	−34.4432	−1.62547	−0.812737	−	0.582631i	$-0.802023\pi$
$449$	−34.4432	−1.62547	−0.812737	+	0.582631i	$0.802023\pi$
$450$	0	0
$451$	−10.2091	−0.480727
$452$	0	0
$453$	20.1469	0.946586
$454$	0	0
$455$	0	0
$456$	0	0
$457$	11.0286	0.515895	0.257948	−	0.966159i	$-0.416954\pi$
$457$	11.0286	0.515895	0.257948	+	0.966159i	$0.416954\pi$
$458$	0	0
$459$	−3.15746	−0.147378
$460$	0	0
$461$	−9.38574	−0.437138	−0.218569	−	0.975822i	$-0.570139\pi$
$461$	−9.38574	−0.437138	−0.218569	+	0.975822i	$0.570139\pi$
$462$	0	0
$463$	2.63515	0.122466	0.0612328	−	0.998124i	$-0.480497\pi$
$463$	2.63515	0.122466	0.0612328	+	0.998124i	$0.480497\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−37.8923	−1.75345	−0.876724	−	0.480994i	$-0.840276\pi$
$467$	−37.8923	−1.75345	−0.876724	+	0.480994i	$0.840276\pi$
$468$	0	0
$469$	15.8470	0.731744
$470$	0	0
$471$	−6.15513	−0.283613
$472$	0	0
$473$	−12.3029	−0.565687
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−0.730286	−0.0334375
$478$	0	0
$479$	−6.31492	−0.288536	−0.144268	−	0.989539i	$-0.546083\pi$
$479$	−6.31492	−0.288536	−0.144268	+	0.989539i	$0.546083\pi$
$480$	0	0
$481$	2.33323	0.106386
$482$	0	0
$483$	−0.417695	−0.0190058
$484$	0	0
$485$	0	0
$486$	0	0
$487$	14.0946	0.638687	0.319343	−	0.947639i	$-0.396538\pi$
$487$	14.0946	0.638687	0.319343	+	0.947639i	$0.396538\pi$
$488$	0	0
$489$	−8.16525	−0.369245
$490$	0	0
$491$	13.2922	0.599869	0.299934	−	0.953960i	$-0.403035\pi$
$491$	13.2922	0.599869	0.299934	+	0.953960i	$0.403035\pi$
$492$	0	0
$493$	19.6430	0.884676
$494$	0	0
$495$	0	0
$496$	0	0
$497$	28.7330	1.28885
$498$	0	0
$499$	37.6503	1.68546	0.842729	−	0.538337i	$-0.180947\pi$
$499$	37.6503	1.68546	0.842729	+	0.538337i	$0.180947\pi$
$500$	0	0
$501$	−10.6401	−0.475365
$502$	0	0
$503$	−0.913200	−0.0407176	−0.0203588	−	0.999793i	$-0.506481\pi$
$503$	−0.913200	−0.0407176	−0.0203588	+	0.999793i	$0.506481\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	37.7488	1.67648
$508$	0	0
$509$	−34.9232	−1.54795	−0.773973	−	0.633219i	$-0.781733\pi$
$509$	−34.9232	−1.54795	−0.773973	+	0.633219i	$0.781733\pi$
$510$	0	0
$511$	−11.6323	−0.514584
$512$	0	0
$513$	−4.78473	−0.211251
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−14.5952	−0.641894
$518$	0	0
$519$	6.66116	0.292393
$520$	0	0
$521$	−28.4317	−1.24562	−0.622809	−	0.782374i	$-0.714009\pi$
$521$	−28.4317	−1.24562	−0.622809	+	0.782374i	$0.714009\pi$
$522$	0	0
$523$	−16.4521	−0.719402	−0.359701	−	0.933068i	$-0.617121\pi$
$523$	−16.4521	−0.719402	−0.359701	+	0.933068i	$0.617121\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1.27170	−0.0553962
$528$	0	0
$529$	−22.9894	−0.999539
$530$	0	0
$531$	−8.39353	−0.364248
$532$	0	0
$533$	54.3112	2.35248
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−1.77316	−0.0765176
$538$	0	0
$539$	12.6390	0.544400
$540$	0	0
$541$	28.4455	1.22297	0.611484	−	0.791257i	$-0.290573\pi$
$541$	28.4455	1.22297	0.611484	+	0.791257i	$0.290573\pi$
$542$	0	0
$543$	16.9027	0.725364
$544$	0	0
$545$	0	0
$546$	0	0
$547$	29.8588	1.27667	0.638334	−	0.769759i	$-0.279624\pi$
$547$	29.8588	1.27667	0.638334	+	0.769759i	$0.279624\pi$
$548$	0	0
$549$	−3.65376	−0.155939
$550$	0	0
$551$	29.7664	1.26809
$552$	0	0
$553$	−50.4696	−2.14618
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−38.0727	−1.61319	−0.806595	−	0.591104i	$-0.798692\pi$
$557$	−38.0727	−1.61319	−0.806595	+	0.591104i	$0.798692\pi$
$558$	0	0
$559$	65.4500	2.76824
$560$	0	0
$561$	−4.22813	−0.178512
$562$	0	0
$563$	−29.4249	−1.24011	−0.620055	−	0.784558i	$-0.712890\pi$
$563$	−29.4249	−1.24011	−0.620055	+	0.784558i	$0.712890\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	4.05444	0.170271
$568$	0	0
$569$	26.8910	1.12733	0.563665	−	0.826003i	$-0.309391\pi$
$569$	26.8910	1.12733	0.563665	+	0.826003i	$0.309391\pi$
$570$	0	0
$571$	−23.4059	−0.979506	−0.489753	−	0.871861i	$-0.662913\pi$
$571$	−23.4059	−0.979506	−0.489753	+	0.871861i	$0.662913\pi$
$572$	0	0
$573$	−8.78473	−0.366987
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−14.8910	−0.619922	−0.309961	−	0.950749i	$-0.600316\pi$
$577$	−14.8910	−0.619922	−0.309961	+	0.950749i	$0.600316\pi$
$578$	0	0
$579$	12.4745	0.518421
$580$	0	0
$581$	16.0908	0.667559
$582$	0	0
$583$	−0.977918	−0.0405012
$584$	0	0
$585$	0	0
$586$	0	0
$587$	23.7896	0.981900	0.490950	−	0.871188i	$-0.336650\pi$
$587$	23.7896	0.981900	0.490950	+	0.871188i	$0.336650\pi$
$588$	0	0
$589$	−1.92710	−0.0794047
$590$	0	0
$591$	−16.2500	−0.668434
$592$	0	0
$593$	−7.93152	−0.325708	−0.162854	−	0.986650i	$-0.552070\pi$
$593$	−7.93152	−0.325708	−0.162854	+	0.986650i	$0.552070\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	15.6334	0.639831
$598$	0	0
$599$	−46.7716	−1.91104	−0.955519	−	0.294931i	$-0.904703\pi$
$599$	−46.7716	−1.91104	−0.955519	+	0.294931i	$0.904703\pi$
$600$	0	0
$601$	−2.18877	−0.0892820	−0.0446410	−	0.999003i	$-0.514214\pi$
$601$	−2.18877	−0.0892820	−0.0446410	+	0.999003i	$0.514214\pi$
$602$	0	0
$603$	3.90854	0.159168
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−20.6382	−0.837678	−0.418839	−	0.908061i	$-0.637563\pi$
$607$	−20.6382	−0.837678	−0.418839	+	0.908061i	$0.637563\pi$
$608$	0	0
$609$	−25.2232	−1.02210
$610$	0	0
$611$	77.6447	3.14117
$612$	0	0
$613$	−45.4674	−1.83641	−0.918205	−	0.396105i	$-0.870362\pi$
$613$	−45.4674	−1.83641	−0.918205	+	0.396105i	$0.870362\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	6.97648	0.280862	0.140431	−	0.990090i	$-0.455151\pi$
$617$	6.97648	0.280862	0.140431	+	0.990090i	$0.455151\pi$
$618$	0	0
$619$	−11.1090	−0.446510	−0.223255	−	0.974760i	$-0.571668\pi$
$619$	−11.1090	−0.446510	−0.223255	+	0.974760i	$0.571668\pi$
$620$	0	0
$621$	−0.103022	−0.00413412
$622$	0	0
$623$	73.2329	2.93401
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−6.40718	−0.255878
$628$	0	0
$629$	1.03415	0.0412342
$630$	0	0
$631$	−0.589056	−0.0234500	−0.0117250	−	0.999931i	$-0.503732\pi$
$631$	−0.589056	−0.0234500	−0.0117250	+	0.999931i	$0.503732\pi$
$632$	0	0
$633$	23.1389	0.919689
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−67.2381	−2.66407
$638$	0	0
$639$	7.08680	0.280349
$640$	0	0
$641$	30.0504	1.18692	0.593460	−	0.804863i	$-0.297761\pi$
$641$	30.0504	1.18692	0.593460	+	0.804863i	$0.297761\pi$
$642$	0	0
$643$	−9.47601	−0.373697	−0.186849	−	0.982389i	$-0.559827\pi$
$643$	−9.47601	−0.373697	−0.186849	+	0.982389i	$0.559827\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−11.8150	−0.464496	−0.232248	−	0.972657i	$-0.574608\pi$
$647$	−11.8150	−0.464496	−0.232248	+	0.972657i	$0.574608\pi$
$648$	0	0
$649$	−11.2397	−0.441196
$650$	0	0
$651$	1.63297	0.0640011
$652$	0	0
$653$	−0.0672925	−0.00263336	−0.00131668	−	0.999999i	$-0.500419\pi$
$653$	−0.0672925	−0.00263336	−0.00131668	+	0.999999i	$0.500419\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−2.86904	−0.111932
$658$	0	0
$659$	21.0565	0.820246	0.410123	−	0.912030i	$-0.365486\pi$
$659$	21.0565	0.820246	0.410123	+	0.912030i	$0.365486\pi$
$660$	0	0
$661$	10.7212	0.417007	0.208503	−	0.978022i	$-0.433141\pi$
$661$	10.7212	0.417007	0.208503	+	0.978022i	$0.433141\pi$
$662$	0	0
$663$	22.4932	0.873563
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0.640911	0.0248162
$668$	0	0
$669$	−16.9651	−0.655907
$670$	0	0
$671$	−4.89272	−0.188881
$672$	0	0
$673$	30.2822	1.16729	0.583646	−	0.812008i	$-0.301626\pi$
$673$	30.2822	1.16729	0.583646	+	0.812008i	$0.301626\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	29.9028	1.14926	0.574630	−	0.818414i	$-0.305146\pi$
$677$	29.9028	1.14926	0.574630	+	0.818414i	$0.305146\pi$
$678$	0	0
$679$	59.3386	2.27721
$680$	0	0
$681$	10.7757	0.412924
$682$	0	0
$683$	8.49293	0.324973	0.162486	−	0.986711i	$-0.448049\pi$
$683$	8.49293	0.324973	0.162486	+	0.986711i	$0.448049\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	18.5696	0.708475
$688$	0	0
$689$	5.20242	0.198196
$690$	0	0
$691$	22.9983	0.874897	0.437449	−	0.899243i	$-0.355882\pi$
$691$	22.9983	0.874897	0.437449	+	0.899243i	$0.355882\pi$
$692$	0	0
$693$	5.42926	0.206241
$694$	0	0
$695$	0	0
$696$	0	0
$697$	24.0722	0.911798
$698$	0	0
$699$	−1.65039	−0.0624236
$700$	0	0
$701$	−10.5497	−0.398456	−0.199228	−	0.979953i	$-0.563844\pi$
$701$	−10.5497	−0.398456	−0.199228	+	0.979953i	$0.563844\pi$
$702$	0	0
$703$	1.56712	0.0591050
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−29.8402	−1.12226
$708$	0	0
$709$	38.7411	1.45495	0.727476	−	0.686134i	$-0.240693\pi$
$709$	38.7411	1.45495	0.727476	+	0.686134i	$0.240693\pi$
$710$	0	0
$711$	−12.4480	−0.466835
$712$	0	0
$713$	−0.0414931	−0.00155393
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−19.0660	−0.712033
$718$	0	0
$719$	−19.9535	−0.744140	−0.372070	−	0.928205i	$-0.621352\pi$
$719$	−19.9535	−0.744140	−0.372070	+	0.928205i	$0.621352\pi$
$720$	0	0
$721$	−15.7337	−0.585952
$722$	0	0
$723$	15.5752	0.579246
$724$	0	0
$725$	0	0
$726$	0	0
$727$	4.98353	0.184829	0.0924144	−	0.995721i	$-0.470542\pi$
$727$	4.98353	0.184829	0.0924144	+	0.995721i	$0.470542\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	29.0091	1.07294
$732$	0	0
$733$	−24.6102	−0.909000	−0.454500	−	0.890747i	$-0.650182\pi$
$733$	−24.6102	−0.909000	−0.454500	+	0.890747i	$0.650182\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	5.23389	0.192793
$738$	0	0
$739$	28.1007	1.03370	0.516850	−	0.856076i	$-0.327104\pi$
$739$	28.1007	1.03370	0.516850	+	0.856076i	$0.327104\pi$
$740$	0	0
$741$	34.0855	1.25216
$742$	0	0
$743$	7.88253	0.289182	0.144591	−	0.989492i	$-0.453813\pi$
$743$	7.88253	0.289182	0.144591	+	0.989492i	$0.453813\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	3.96869	0.145207
$748$	0	0
$749$	−38.0998	−1.39214
$750$	0	0
$751$	21.3763	0.780031	0.390015	−	0.920808i	$-0.372470\pi$
$751$	21.3763	0.780031	0.390015	+	0.920808i	$0.372470\pi$
$752$	0	0
$753$	−18.2130	−0.663717
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−17.6845	−0.642755	−0.321377	−	0.946951i	$-0.604146\pi$
$757$	−17.6845	−0.642755	−0.321377	+	0.946951i	$0.604146\pi$
$758$	0	0
$759$	−0.137955	−0.00500746
$760$	0	0
$761$	−30.5834	−1.10865	−0.554323	−	0.832302i	$-0.687023\pi$
$761$	−30.5834	−1.10865	−0.554323	+	0.832302i	$0.687023\pi$
$762$	0	0
$763$	−19.8408	−0.718287
$764$	0	0
$765$	0	0
$766$	0	0
$767$	59.7940	2.15904
$768$	0	0
$769$	8.39447	0.302712	0.151356	−	0.988479i	$-0.451636\pi$
$769$	8.39447	0.302712	0.151356	+	0.988479i	$0.451636\pi$
$770$	0	0
$771$	14.8505	0.534827
$772$	0	0
$773$	42.8082	1.53970	0.769851	−	0.638223i	$-0.220330\pi$
$773$	42.8082	1.53970	0.769851	+	0.638223i	$0.220330\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−1.32793	−0.0476393
$778$	0	0
$779$	36.4782	1.30697
$780$	0	0
$781$	9.48986	0.339574
$782$	0	0
$783$	−6.22113	−0.222325
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−10.3286	−0.368174	−0.184087	−	0.982910i	$-0.558933\pi$
$787$	−10.3286	−0.368174	−0.184087	+	0.982910i	$0.558933\pi$
$788$	0	0
$789$	−28.2291	−1.00498
$790$	0	0
$791$	−68.2913	−2.42816
$792$	0	0
$793$	26.0287	0.924308
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−2.84333	−0.100716	−0.0503580	−	0.998731i	$-0.516036\pi$
$797$	−2.84333	−0.100716	−0.0503580	+	0.998731i	$0.516036\pi$
$798$	0	0
$799$	34.4142	1.21749
$800$	0	0
$801$	18.0624	0.638203
$802$	0	0
$803$	−3.84190	−0.135578
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−2.33130	−0.0820657
$808$	0	0
$809$	17.3440	0.609783	0.304891	−	0.952387i	$-0.401380\pi$
$809$	17.3440	0.609783	0.304891	+	0.952387i	$0.401380\pi$
$810$	0	0
$811$	11.8281	0.415341	0.207670	−	0.978199i	$-0.433412\pi$
$811$	11.8281	0.415341	0.207670	+	0.978199i	$0.433412\pi$
$812$	0	0
$813$	9.84503	0.345280
$814$	0	0
$815$	0	0
$816$	0	0
$817$	43.9596	1.53795
$818$	0	0
$819$	−28.8831	−1.00926
$820$	0	0
$821$	−19.9201	−0.695217	−0.347608	−	0.937640i	$-0.613006\pi$
$821$	−19.9201	−0.695217	−0.347608	+	0.937640i	$0.613006\pi$
$822$	0	0
$823$	−37.0022	−1.28982	−0.644909	−	0.764260i	$-0.723105\pi$
$823$	−37.0022	−1.28982	−0.644909	+	0.764260i	$0.723105\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	3.61973	0.125870	0.0629351	−	0.998018i	$-0.479954\pi$
$827$	3.61973	0.125870	0.0629351	+	0.998018i	$0.479954\pi$
$828$	0	0
$829$	−42.3796	−1.47191	−0.735953	−	0.677033i	$-0.763266\pi$
$829$	−42.3796	−1.47191	−0.735953	+	0.677033i	$0.763266\pi$
$830$	0	0
$831$	−24.4616	−0.848564
$832$	0	0
$833$	−29.8017	−1.03257
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0.402761	0.0139214
$838$	0	0
$839$	31.6705	1.09339	0.546694	−	0.837332i	$-0.315886\pi$
$839$	31.6705	1.09339	0.546694	+	0.837332i	$0.315886\pi$
$840$	0	0
$841$	9.70250	0.334569
$842$	0	0
$843$	15.6447	0.538832
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−37.3286	−1.28263
$848$	0	0
$849$	9.82969	0.337354
$850$	0	0
$851$	0.0337422	0.00115667
$852$	0	0
$853$	−10.7240	−0.367184	−0.183592	−	0.983003i	$-0.558773\pi$
$853$	−10.7240	−0.367184	−0.183592	+	0.983003i	$0.558773\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	11.8384	0.404391	0.202196	−	0.979345i	$-0.435192\pi$
$857$	11.8384	0.404391	0.202196	+	0.979345i	$0.435192\pi$
$858$	0	0
$859$	8.53452	0.291194	0.145597	−	0.989344i	$-0.453490\pi$
$859$	8.53452	0.291194	0.145597	+	0.989344i	$0.453490\pi$
$860$	0	0
$861$	−30.9106	−1.05343
$862$	0	0
$863$	−38.1498	−1.29864	−0.649318	−	0.760517i	$-0.724945\pi$
$863$	−38.1498	−1.29864	−0.649318	+	0.760517i	$0.724945\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−7.03043	−0.238766
$868$	0	0
$869$	−16.6689	−0.565455
$870$	0	0
$871$	−27.8437	−0.943449
$872$	0	0
$873$	14.6355	0.495335
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−2.01003	−0.0678737	−0.0339369	−	0.999424i	$-0.510805\pi$
$877$	−2.01003	−0.0678737	−0.0339369	+	0.999424i	$0.510805\pi$
$878$	0	0
$879$	−17.3449	−0.585031
$880$	0	0
$881$	2.16862	0.0730627	0.0365313	−	0.999333i	$-0.488369\pi$
$881$	2.16862	0.0730627	0.0365313	+	0.999333i	$0.488369\pi$
$882$	0	0
$883$	55.0263	1.85178	0.925891	−	0.377792i	$-0.123317\pi$
$883$	55.0263	1.85178	0.925891	+	0.377792i	$0.123317\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−50.3100	−1.68925	−0.844623	−	0.535362i	$-0.820175\pi$
$887$	−50.3100	−1.68925	−0.844623	+	0.535362i	$0.820175\pi$
$888$	0	0
$889$	69.6192	2.33495
$890$	0	0
$891$	1.33909	0.0448612
$892$	0	0
$893$	52.1502	1.74514
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0.733907	0.0245045
$898$	0	0
$899$	−2.50563	−0.0835673
$900$	0	0
$901$	2.30585	0.0768190
$902$	0	0
$903$	−37.2501	−1.23961
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−13.1238	−0.435769	−0.217885	−	0.975975i	$-0.569916\pi$
$907$	−13.1238	−0.435769	−0.217885	+	0.975975i	$0.569916\pi$
$908$	0	0
$909$	−7.35988	−0.244112
$910$	0	0
$911$	−33.8534	−1.12161	−0.560806	−	0.827947i	$-0.689509\pi$
$911$	−33.8534	−1.12161	−0.560806	+	0.827947i	$0.689509\pi$
$912$	0	0
$913$	5.31443	0.175882
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−25.7304	−0.849694
$918$	0	0
$919$	−0.0908103	−0.00299556	−0.00149778	−	0.999999i	$-0.500477\pi$
$919$	−0.0908103	−0.00299556	−0.00149778	+	0.999999i	$0.500477\pi$
$920$	0	0
$921$	−22.1862	−0.731060
$922$	0	0
$923$	−50.4851	−1.66174
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−3.88060	−0.127456
$928$	0	0
$929$	−22.6510	−0.743156	−0.371578	−	0.928402i	$-0.621183\pi$
$929$	−22.6510	−0.743156	−0.371578	+	0.928402i	$0.621183\pi$
$930$	0	0
$931$	−45.1606	−1.48008
$932$	0	0
$933$	19.9407	0.652828
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−9.99276	−0.326449	−0.163225	−	0.986589i	$-0.552190\pi$
$937$	−9.99276	−0.326449	−0.163225	+	0.986589i	$0.552190\pi$
$938$	0	0
$939$	10.9945	0.358793
$940$	0	0
$941$	−28.6412	−0.933675	−0.466838	−	0.884343i	$-0.654607\pi$
$941$	−28.6412	−0.933675	−0.466838	+	0.884343i	$0.654607\pi$
$942$	0	0
$943$	0.785426	0.0255770
$944$	0	0
$945$	0	0
$946$	0	0
$947$	8.22217	0.267185	0.133592	−	0.991036i	$-0.457349\pi$
$947$	8.22217	0.267185	0.133592	+	0.991036i	$0.457349\pi$
$948$	0	0
$949$	20.4385	0.663461
$950$	0	0
$951$	24.6249	0.798518
$952$	0	0
$953$	48.8140	1.58124	0.790620	−	0.612307i	$-0.209759\pi$
$953$	48.8140	1.58124	0.790620	+	0.612307i	$0.209759\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−8.33066	−0.269292
$958$	0	0
$959$	76.8784	2.48253
$960$	0	0
$961$	−30.8378	−0.994767
$962$	0	0
$963$	−9.39706	−0.302816
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−17.4141	−0.559999	−0.279999	−	0.960000i	$-0.590334\pi$
$967$	−17.4141	−0.559999	−0.279999	+	0.960000i	$0.590334\pi$
$968$	0	0
$969$	15.1076	0.485326
$970$	0	0
$971$	−0.795340	−0.0255237	−0.0127618	−	0.999919i	$-0.504062\pi$
$971$	−0.795340	−0.0255237	−0.0127618	+	0.999919i	$0.504062\pi$
$972$	0	0
$973$	61.3893	1.96805
$974$	0	0
$975$	0	0
$976$	0	0
$977$	47.2594	1.51196	0.755982	−	0.654593i	$-0.227160\pi$
$977$	47.2594	1.51196	0.755982	+	0.654593i	$0.227160\pi$
$978$	0	0
$979$	24.1872	0.773025
$980$	0	0
$981$	−4.89361	−0.156241
$982$	0	0
$983$	18.3638	0.585715	0.292857	−	0.956156i	$-0.405394\pi$
$983$	18.3638	0.585715	0.292857	+	0.956156i	$0.405394\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−44.1906	−1.40660
$988$	0	0
$989$	0.946510	0.0300973
$990$	0	0
$991$	55.5141	1.76346	0.881732	−	0.471750i	$-0.156378\pi$
$991$	55.5141	1.76346	0.881732	+	0.471750i	$0.156378\pi$
$992$	0	0
$993$	−5.98481	−0.189922
$994$	0	0
$995$	0	0
$996$	0	0
$997$	13.2769	0.420482	0.210241	−	0.977650i	$-0.432575\pi$
$997$	13.2769	0.420482	0.210241	+	0.977650i	$0.432575\pi$
$998$	0	0
$999$	−0.327525	−0.0103624

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6000.2.a.bi.1.4		4
4.3	odd	2		3000.2.a.k.1.1	✓	4
5.2	odd	4		6000.2.f.s.1249.4		8
5.3	odd	4		6000.2.f.s.1249.5		8
5.4	even	2		6000.2.a.bf.1.1		4
12.11	even	2		9000.2.a.u.1.1		4
20.3	even	4		3000.2.f.e.1249.4		8
20.7	even	4		3000.2.f.e.1249.5		8
20.19	odd	2		3000.2.a.n.1.4	yes	4
60.59	even	2		9000.2.a.x.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3000.2.a.k.1.1	✓	4	4.3	odd	2
3000.2.a.n.1.4	yes	4	20.19	odd	2
3000.2.f.e.1249.4		8	20.3	even	4
3000.2.f.e.1249.5		8	20.7	even	4
6000.2.a.bf.1.1		4	5.4	even	2
6000.2.a.bi.1.4		4	1.1	even	1	trivial
6000.2.f.s.1249.4		8	5.2	odd	4
6000.2.f.s.1249.5		8	5.3	odd	4
9000.2.a.u.1.1		4	12.11	even	2
9000.2.a.x.1.4		4	60.59	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 6000 = 2^{4} \cdot 3 \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6000.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +4.05444 q^{7} +1.00000 q^{9} +1.33909 q^{11} -7.12382 q^{13} -3.15746 q^{17} -4.78473 q^{19} +4.05444 q^{21} -0.103022 q^{23} +1.00000 q^{27} -6.22113 q^{29} +0.402761 q^{31} +1.33909 q^{33} -0.327525 q^{37} -7.12382 q^{39} -7.62389 q^{41} -9.18749 q^{43} -10.8993 q^{47} +9.43849 q^{49} -3.15746 q^{51} -0.730286 q^{53} -4.78473 q^{57} -8.39353 q^{59} -3.65376 q^{61} +4.05444 q^{63} +3.90854 q^{67} -0.103022 q^{69} +7.08680 q^{71} -2.86904 q^{73} +5.42926 q^{77} -12.4480 q^{79} +1.00000 q^{81} +3.96869 q^{83} -6.22113 q^{87} +18.0624 q^{89} -28.8831 q^{91} +0.402761 q^{93} +14.6355 q^{97} +1.33909 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} + q^{7} + 4 q^{9} - q^{11} - 11 q^{13} - 8 q^{19} + q^{21} - 3 q^{23} + 4 q^{27} - 3 q^{29} - 14 q^{31} - q^{33} - 21 q^{37} - 11 q^{39} + 7 q^{41} - 10 q^{43} - 9 q^{47} + 15 q^{49} - 7 q^{53}+ \cdots - q^{99}+O(q^{100}) \) 4 * q + 4 * q^3 + q^7 + 4 * q^9 - q^11 - 11 * q^13 - 8 * q^19 + q^21 - 3 * q^23 + 4 * q^27 - 3 * q^29 - 14 * q^31 - q^33 - 21 * q^37 - 11 * q^39 + 7 * q^41 - 10 * q^43 - 9 * q^47 + 15 * q^49 - 7 * q^53 - 8 * q^57 - 12 * q^59 - 3 * q^61 + q^63 - 13 * q^67 - 3 * q^69 - 33 * q^71 - 11 * q^73 - 7 * q^77 - 13 * q^79 + 4 * q^81 - 21 * q^83 - 3 * q^87 + 12 * q^89 - 17 * q^91 - 14 * q^93 - 5 * q^97 - q^99

Introduction

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